Sudoku Diagonal Rules: When the Diagonals Count

In classic Sudoku there is no diagonal rule at all. Rows, columns and 3×3 boxes are the whole of it - a diagonal may carry the same digit twice, three times or not at all, and nothing about the puzzle objects.

The rule people are looking for belongs to a variant: Sudoku X, also sold as Diagonal Sudoku or X-Sudoku. There the two main diagonals are units in their own right, exactly like a row. If a puzzle's instructions mention diagonals at all, that is the puzzle you are holding.

The rule, in one sentence

Every classic rule still applies, and each of the two corner-to-corner diagonals must also contain the digits 1 to 9 exactly once. Nothing is taken away; one constraint is added, twice.

Which cells the rule actually touches

Seventeen of the eighty-one, not eighteen: nine cells on each diagonal, and the centre cell belongs to both. That single overlap is the most constrained square on the whole board.

• An ordinary cell answers to three units - its row, its column, its box.
• A cell on one diagonal answers to four.
• The centre cell answers to five: row, column, box and both diagonals.
• The remaining sixty-four cells are untouched. On those, the puzzle is exactly the Sudoku you already know.

A diagonal does add up to 45, because it holds 1 to 9 once each - but that is a consequence of the rule, not the rule itself. Nothing here asks you to add anything up. Sums belong to a different variant, Killer Sudoku, where cages carry target totals.

What the diagonal rule is not

• Only the two main diagonals count. The short diagonals, the broken ones that wrap around an edge, and the diagonals inside a single 3×3 box are all unconstrained.
• It does not change anything else. Rows, columns and boxes work as before, and the puzzle still has exactly one solution reachable by logic.
• It is not a rule you can apply to a classic puzzle. Forcing 1 to 9 onto the diagonals of a normal Sudoku will usually contradict its solution, because the setter never promised it.

A move only the diagonal can make

The rule earns its keep by placing digits that classic logic cannot reach. On the board above, look at the main diagonal, running from the top-left corner to the bottom-right.

The diagonal already carries a 4, a 7, a 6, an 8 and a 2, so it still needs a 3 somewhere among its four empty cells. Take them one at a time:

• Top-left corner: its row already has the 3 near the right-hand end.
• The cell just below it: its column carries a 3 at the very bottom of the grid.
• The cell in the bottom-right box: its column is the one the top-row 3 sits in.
• The centre: nothing rules it out.

So the 3 goes in the centre. Now notice what happened, because it is the whole point of the variant. Ask the cell what fits and you get nowhere - row, column and box leave the centre with three live candidates, 1, 3 and 9, and a classic solver moves on. Ask the diagonal where its 3 can live and the answer is forced. This is an ordinary hidden single with the diagonal as the unit, and it is the single most productive habit in Sudoku X.

The corollary is worth saying plainly: scan the diagonals the way you scan rows and columns. Two extra units mean two extra places to hunt, and beginners lose most of their time in this variant by forgetting they are there. The scanning itself is the same crosshatching you already use.

Does the extra rule make it harder?

Not the way people expect. A constraint takes freedom away from the setter as well as from the solver: with two more units to satisfy, each digit you place tells you more than it would on a classic board, so deductions come faster once you look in the right places. Sudoku X plays differently rather than strictly tougher - which is why we score it at 1.2× a classic board of the same level rather than treating it as a difficulty step.

The one genuinely awkward square is the centre. Five units meet there, so it is usually among the last cells to open up - and among the most useful once it does.

Other puzzles that put rules on diagonals

• Anti-Diagonal Sudoku - the mirror image: each main diagonal may hold only three distinct digits, each appearing three times.
• Argyle Sudoku - a lattice of marked diagonals crosses the grid, and no digit may repeat along any of them.
• Sujiken - a Sudoku cut in half along its diagonal; in the 45-cell triangle that remains, diagonals are units too.

Where to play it

Sudoku X is one of the types you can play here, and the fastest way to learn a rule is to be held to it. The live board shades both diagonals so you cannot forget where they run, and it checks every digit you enter against the puzzle's one solution - so a digit that breaks a diagonal is flagged the moment you write it, no differently from one that breaks a row. Start at Easy while the two extra lines are still something you have to remember to look at, then move up.

The full Sudoku X guide goes further into the diagonal-only techniques: diagonal pairs, pointing from a diagonal into a box, and what to do with that five-unit centre cell. If you would rather solve on paper, the printable generator prints Sudoku X sheets with the diagonals marked.